Abstract.While Rudolph Schindler's "space reference frame"
is becoming better known, its relationship to the "row"
has only been recently investigated. The theory of the "row"
counters traditional proportional notions, many of which are derived
from the principle of geometric similitude: a principle which
is mostly represented in architectural drawings by regulating
lines and triangulation. Here, Lionel March presents the simple
mathematics of row theory. A short background note concludes the
paper.

INTRODUCTIONThe approach to architectural dimensioning
taken by Rudolph Schindler (1887-1953) is described by Jin-Ho
Park in a complimentary paper in this issue of the Nexus
Network Journal [2003]. While Schindler's "space reference
frame" is becoming better known [Schindler 1946], its relationship
to the "row" is only to be found in Park's recent investigation.
The theory of the "row" counters traditional proportional
notions, many of which are derived from the principle of geometric
similitude: a principle which is mostly represented in architectural
drawings by regulating lines and triangulation. Park gives examples
of this. Here, the simple mathematics of row theory is presented.
A short background note concludes the paper.

As an architect, Schindler had to choose a suitable module
for his "unit." In most of his designs, he chose 48
in. This module might be divided, Schindler argued, by 1/2, 1/3,
1/4, without losing a "feeling" for the dimensions.
In fact, Schindler used 36 in., 24 in. and 12 in. as refinements
in plan, and the one third divisions, 16 in. and 32 in., almost
exclusively in elevation.

Consider a general module M. The equivalent divisions
for dimensional refinements are then M/4, M/3,
M/2, 2M/3, 3M/4; but clearly this could
be further generalized.

INTRODUCING THE ROW CONCEPTIn his unpublished
1917 Emma Church School lecture notes, Schindler cryptically
defined a row as a "following of unequal units with definite
changes [of] relation." He then gave this example:

1/2, 2/3, 3/4, 4/5, 5/6, ...

After Robinson [1899], he suggested an interpretation for
this sequence, or row, as intervallic ratios in a musical scale.
Schindler suggested the general rule. If p/q is a term in a row,
then the successor term is given by the rule:

while Robinson, in addition, set out the rows beginning with
1/4 and 1/5.

The fractional notation, p/q, was used by both
Robinson and Schindler. Each row may be depicted as a sequence
of rectangles by setting p/q as the vertex (p,
q), and the other vertices as (p, 0), (0, q),
(0, 0). This produces a diagram which the classical Greeks described
as gnomonic (Fig. 1).

Every second term, read as a fraction, may be reduced to its
coprime form:

1/3,

3/5,

5/7,

7/9,

...

1/2,

2/3,

3/4,

4/5,

...

These second terms belong to row (1), while the residual terms
belong to a new kind of row in which the successor rule is

Treating p/q as a reducible fraction leads to
complications. The notation lends itself to misinterpretations.
Take as an example the term 9/19 given by Robinson and cited
by Schindler. Its predecessor is 8/18, and its successor is 10/20.
Suppose the terms represent the dimensions of a window, as Schindler
imagines. It would be conventional to speak about the proportions
of the window in terms of the ratios 4 : 9, 9 : 19, and 1 : 2.
If the fractional terms, 8/18 and 10/20, are reduced to 4/9 and
1/2 respectively, then it will be seen that the predecessor comes
from row (5) and the successor from row (1), whereas the original
term 9/19 belongs to row (10).

To avoid such multiple interpretations in the remaining sections
of this paper, the notation pxq will be used, where x is read
as "by". This notation conforms to the architectural
convention of dimensioning rooms, and corresponds to the rectangles
in the gnomonic diagram, Fig. 1.

THE ELEMENTARY THEORY OF ROWSThe row arises in a
limited architectural context. A term pxq is to be understood as the dimensions of a rectangular
space be it a room, a window opening, or a door. To remind us
of this spatial usage we call the term pxq, a raum, with p designated as the width
and q as the length.

If is
a raum in a row, then the first raum of the row is 1 x
(q-p+1). The kth raum is thenk
x (q - p + k).
Call this the row (q - p). Any nth row may
be represented abstractly as the infinite set:

row (n) = { kx
(n + k) | k = 1, 2, ... }.

A designer may consider a space, or raum pxq. Within a space reference frame, the designer may enlarge,
or reduce, the space by one module in one of two direction:

(p + 1) x
(q + 1)

(p + 1) xq

(p + 1) x
(q - 1)

px (q
+ 1)

pxq

px (q
- 1)

(p - 1) x
(q + 1)

(p - 1) xq

(p - 1) x
(q - 1)

In doing so the designer will move to neighboring rows:

row(q - p)

row(q - p - 1)

row(q - p - 2)

row(q - p + 1)

row(q - p)

row(q - p - 1)

row(q - p + 2)

row(q - p + 1)

row(q - p)

To generalize this; consider that p is changed by ±
a and all integers in between including 0, and q
by ± b similarly. Providing widths and lengths
are preserved, this gives (2a +1)(2b + 1) related
raum including the original:

(p + a)

x

(q + b)

...

(p + a)

x

q

...

(p + a)

x

(q - b)

:

:

:

p

x

(q + b)

...

p

x

q

...

p

x

(q - b)

:

:

:

(p - a)

x

(q + b)

...

(p - a)

x

q

...

(p - a)

x

(q - b)

ranging from row(q - p - a - b)
to row(q - p + a + b) and all rows
in between.

Fig. 2. Illustration of the first
seven rows depicted as rectangles. The raum 5 x 8 is shown black.
The outlined rectangles show the range resulting from adjustments
to the width a = 2, 1, 0, -1, -2, and of the length b
= 1, 0, -1. There are (2.2 + 1)(2.1 + 1) = 15 neighborly rectangles,
including the original rectangle.

Except for row(1) the rows include rectangles in which the
two dimensions are not coprime. That is, they share a common
factor. Thus in row(2), the rectangle 2x4
has the same ratio as the rectangle 1x2
in row(1). It is useful to extract the coprime rectangles in
which the two dimensions are prime to one another as shown in
Table 1.

Table 1. Examples of
coprime raum where the two dimensions are prime to one another.

Consider the simple row(n) commencing 1xn.
The difference is n - 1. Suppose n - 1 has a prime
factor u. Let n - 1 = uv. The kth
raum in the row is (k, n + k - 1) = (k,
k + uv). If k is a multiple of u,
say uw, then the kth term may be written (uw,
uw + uv) = (uv, u (v+w)).
Hence if the difference has a prime factor u, every uth
term will be a non-prime term. This will be true for all prime
factors of the difference. Eliminating all non-coprime terms
will leave just the terms of the coprime row. All terms of coprime
rows are based on a single unit module. The non-coprime terms
from the simple rows are based on multiple units according the
their positions in the simple row.

This may be seen in an example using a graphic representation
of anthyphairesis [Fowler 1999; March 1999a, b]. Simply put,
this means subtracting successively the largest squares from
a given rectangle and residuals until the procedure stops. The
procedure will not stop if the rectangle is incommensurable,
but this is not the case here. For example, consider the partial
diagram of row(6): 1x7, ... 11x17 (Fig. 3).

Fig. 3. Partial gnomonic diagram of row(6).

Successive subtraction of squares in these rectangles reveals
the natural unit of each. Fig. 4 illustrates both the coprime
and non-coprime rectangles of row(6).

It can be seen that a particular row may contain raum proportional
to raum in other rows, as in this example. The question may be
asked: when does a raum have the proportion of the root ratio
r : s, that is a ratio in its coprime form? Consider
the kth raum kx (n
+ k) from row(n). The required condition will be
satisfied if r / s = k / (n + k),
or if k = r n / (s - r). Now, k
is an integer so that (s-r) must be a factor of
r n for the condition to be met. This requires

r n modulo (s-r ) = 0.

If (s - r) = 1, that is r / s
is subsuperparticular, or rx
s is a raum in row(1), it will be seen that every row contains
a raum proportional to r : s. That is, every row
contains one raum proportional to 1 : 2, the double square, 2
: 3, 3 : 4, 4 : 5, ... .

Fig. 4. The first twelve raum of row(6).
Coprime rectangles are above the line. Each exhibits a defining
1x1
square module. Below the line are non-coprime rectangles exhibiting
multiple square modules. Top, rectangles defined by a 2x2 module; center,
rectangles defined by a 3x3 module; bottom, rectangles defined by a 6x6 module.

According to Schindler, the unit of dimension, or module,
is the choice of the architect. "He needs a unit dimension
which is large enough to give his building scale, rhythm and
cohesion. And last, but most important for the 'space architect,'
it must be a unit which he can carry palpably in his mind in
order to be able to deal with space forms freely but accurately
in his imagination" [Schindler 1946; see Park 2003].

Robinson [1899: 298] gives an example of dimensional change
using the method of regulating lines. He states that the "fundamental
idea of proportion ... is that all parts share the same general
character - what geometricians call "similar"; that
is, that if one part is seven wide and ten high another part
that is only eight high shall be about, or exactly, five and
six tenths wide" (Fig. 5).

This represents a linear scale change of 80%; but Robinson
does not take the opportunity to point this out.

Fig. 6. The geometric method gives
rise to scale changes such as this 80% linear reduction in the
unit module.

In Fig. 6, the method of anthyphairesis, successive subtraction,
reveals the "natural" unit of measure, or module of
a rectangle [March 1999a, b]. This so called "geometric"
method leads to two problems for Schindler. First, there is the
question of uncertain scale and choice of unit. Second, there
is the potential for unacceptable computation "for easy
grasp." Schindler sees no good reason to accept a fraction
such as 3/5. If, in figure 6, the unit is presumed to be 1 ft.,
then the reduced rectangle would have the dimensions 8 ft. by
5 ft. 7 1/5 in. which "it is hardly possible to visualize."

By using the 48 inch unit, Schindler proposed that it is possible
to "feel" the larger rectangle as 1 3/4 x 2 1/2 units,
and the smaller as either 1 1/2 x 2 units, or 1 1/3 x 2 units.
This degree of refinement of the dimensions allows the architect
"to carry in his mind" the design concept.

MODULAR COORDINATIONSchindler corresponded with the Chairman,
Frederick Heath Jr., of the U.S. Producers' Council, Subcommittee
on Modular Products, from 1944. His then unpublished "Reference
Frames in Space" [1946] was duplicated and circulated to
the Subcommittee on Building Layouts. The meeting which considered
Schindler's paper agreed on the term "modular coordination"
for product standardization. In a letter to Schindler, Heath
wrote:

It is recognized that through your years of experience you
have made a substantial contribution to the art of modular planning.
You refer to the 4 in. module as 'texture' and the 4 ft. as 'rhythm'.
That expression was helpful to the Committee, and there was general
agreement that we are dealing with texture rather than rhythm.

Schindler's paper was positively reviewed in Moduletter
38, the official notice of the Subcommittee.

An illustration will demonstrate the problem of employing
the principle of geometric similitude with a modular grid. Suppose
a ratio of 3 : 5 is adopted. The first modular rectangle will
be 3M x 5M, where M is the chosen module.
If the length is extended to 4M, the width will fall off
the grid. The same is true for lengths of 5M, 6M,
7M, 8M, and 9MM. Only when the length 10M
is reached does the width 6M fall on the grid. It is easy
to see that only rectangles 3kM x 5kM, for integer
values of k, fall wholly on the grid (Fig. 7).

This example employs a ratio, 3 : 5, that Schindler recognized
as a rational approximation to the golden section, although he
rejected the value of the section in architectural design. Nevertheless,
the example serves to introduce Le Modulor in which this ratio
appears in the Fibonacci sequence of proportions promoted by
Le Corbusier (Figure 8):

Figure 8. A representation of Le Corbusier's
Le Modulor grid.

It comes as no surprise that Le Corbusier's proportional palette
is a subset of Schindler's universal set {row(k) | k
= 0, 1, 2, ... }. It may be of marginal interest that the first
sequence -- from top to bottom -- in Figure 8 is row(0); that
the next diagonal of rectangles come from row(1), row(1), row(2),
row(3), row(5), ...; the next from row(2), row(3), row(5), row(8),
...; the next from row(4), row(6), row(10), ...; then row(7),
row(11); and finally row(12). If the diagram is extended it will
be found that the raum along the diagonals all come from Fibonacci
sequences of rows:

Rows

(1), (1), (2), (3), (5), (8), (13), ...

(2), (3), (5), (8), (13), (21), (34), ...

(4), (6), (10), (16), (26), (42), ....

(7), (11), (18), (29), (47), (76), ...

(12), (19), (31), (50), (81), ...

Schindler sought freedom from these artificial constraints
and limitations. Proportion as such was not the issue. He was
more concerned with the preservation of scale throughout a work,
the rhythmic relationships and the play of the unit system. Where
necessary the system could be broken. Consistency was no virtue.
The grid did not have to be square. "It is not necessary
that the designer be completely enslaved by the grid. I have
found that occasionally a space-form may be improved by deviating
slightly from the unit. Such sparing deviation does not invalidate
the system as a whole but merely reveals the limits inherent
in all mechanical schemes" [Schindler 1946].

IMPERIAL AND METRICSchindler,
like his exact contemporary Le Corbusier, was no enthusiast of
the metric system for architectural purposes. In the correspondence
with the Producers' Council, Heath sent Schindler a paper proposing
a 40 in. unit module arguing that within building tolerances
it was equivalent to a meter. For a quarter of a century, Schindler
had used 48 in. with few exceptions. The 40 in. module may be
divided by six factors 2, 4, 5, 8, 10, 20: the 48 in. module
by eight factor 2, 3, 4, 6, 8, 12, 16, 24. The metric equivalent
of 40 in. is 100 cm. which can be divided by the seven factors
2, 4, 5, 10, 20, 25, 50. For practical purposes these metric
measures include 2 in., 4 in., 8 in., 10 in., and 20 in. Imperial
equivalents. The metric equivalent to 48 in. is 120 cm. Now 120
is the ninth Ramanujan "highly composite" number defined
as a number that counting from 1 sets a record for the number
of its divisors. 120 is the first number with 16 divisors including
1 and 120. The factors can be represented on a modular lattice,
Fig. 9.

BACKGROUNDMy interest in modular coordination
and proportional systems goes back to my collaboration
with Philip Steadman, who was largely responsible for the chapters
"Modules and numbers" and
"Proportion and series" in The Geometry of Environment
[March and Steadman 1971: 199-241].
In [March 1981], I wrote a short note on a certain class of tartan
grids which I had made use of
in previous design work.

I have written about Schindler's dimensioning system in several
papers [March 1993a, b, c; 1999a]. I was unaware of Schindler's
lecture notes at the time of writing. Judith Sheine first pointed
out to me the sequence 1/2, 2/3. 3/4.4/5, ... and the musical
scale which she had come across in Schindler's notes when researching
her monograph [Sheine, 2001]. She believed it confirmed my emphasis
on the musical analogy in Schindler's work. In the process of
his doctoral investigation, Jin-Ho Park came across the same
notebook and made the valuable connection with Robinson's articles
[1898-99]. This source material has revised my views. I am no
longer so convinced by the musical analogy in the form that I
first promoted it. Any ratio involving small numbers will reproduce
the same ratios to be found in musical theory. I was aware of
this [1993a: 94-5]. Schindler's interest in Robinson's paper
on proportion shown in his lecture notes suggests a more liberal
appreciation of architectural proportions than adherence to musical
harmony would permit. His approach is best explained arithmetically
rather than geometrically, and his lecture notes confirm this.

I previously suggested that Schindler's system of dimensioning
was "classical," and I believe this to be true. His
son, Mark Schindler, remembers that his father would often relax
in the evenings by examining "classical" architectural
drawings. Schindler's lecture notes show a broad interest in
architectural history. Where he introduces the notion of a modular
unit and a "net for planning" based on the "largest
common division," he sidebars ten dimensions of the thirteenth
century Elisabethkirche, Marburg, as multiples of a 17 ft. unit.
On the very next page he describes the concept of a "row."
Again, it was Jin-Ho Park who brought the Schindler-Heath correspondence
on modular coordination to my attention.

In Architectonics of Humanism [March 1998], I explained
the classical understanding and application of ratios and proportion.
The Greek gnomonic diagrams were illustrated there. Indeed, before
I knew about Schindler's rows, I had speculated that certain
temple platforms may have been derived by gnomonic methods. The
platform of the Parthenon may be thought of as being generated
from a 1x10 rectangle by gnomonic additions.
That this base rectangle contains both the divine monad, 1, and
the Pythagorean decad, 10, is surely no coincidence. These observation
support the points made by Hendrik Berlage cited in Park [2003],
Figure 10. In subsequent papers [March 1999b, c] I took up David
Fowler's reconstruction of the mathematics associated with Plato's
academy [1999], and related it to the pictorial methods characteristic
of modern "shape grammars." This is where I employed
the anthyphairetic representation of ratios.

Figure 10. The platform of the Parthenon
seen as a gnomonic diagram generated from a 1x 10 rectangle, tinted.
The gnomonic rectangles belong to Schindler's row(9).

ABOUT THE AUTHOROn the personal recommendation
of Alan Turing, Lionel
March was admitted to Magdalene College, Cambridge, to
read mathematics under Dennis Babbage. There he gained a first
class degree in mathematics and architecture while taking an
active part in Cambridge theater life. In the early sixties,
he was awarded an Harkness Fellowship of the Commonwealth Fund
at the Joint Center for Urban Studies, Harvard University and
Massachusetts Institute of Technology under the directorships
of Martin Meyerson and James Q Wilson. He returned to Cambridge
and joined Sir Leslie Martin and Sir Colin Buchanan in preparing
a plan for a national and government center for Whitehall. He
was the first Director of the Centre for Land Use and Built Form
Studies, now the Martin Centre for Architectural and Urban Studies,
Cambridge University. As founding Chairman of the Board of the
private computer-aided design company, he and his colleagues
were among the first contributors to the 'Cambridge Phenomenon'
- the dissemination of Cambridge scholarship into high-tech industries.
In 1978, he was awarded the Doctor of Science degree for mathematical
and computational studies related to contemporary architectural
and urban problems.
Before coming to Los Angeles he was Rector and Vice-Provost of
the Royal College of Art, London. During his Rectorship he served
as a Governor of Imperial College of Science and Technology.
He has held full Professorships in Systems Engineering at the
University of Waterloo, Ontario; and in Design Technology at
The Open University, Milton Keynes. At The Open University, as
Chair, he doubled the faculty in Design and established the Centre
for Configurational Studies. He came to UCLA in 1984 as a Professor
in the Graduate School of Architecture and Urban Planning. He
was Chair of Architecture and Urban Design from 1985-91. He is
currently Professor in Design and Computation and a member of
the Center for Medieval and Renaissance Studies. He was a member
of UCLA's Council on Academic Personnel from 1993, and its Chair
for 1995/6. He is a General Editor of Cambridge Architectural
and Urban Studies, and Founding Editor of the journal Planning
and Design. The journal is one of four sections of Environment
and Planning, which stands at "the top of the citation
indexes." Among the books he has authored and edited are:
The Geometry of Environment, Urban Space and Structures,
The Architecture of Form, and R. M. Schindler: Composition
and Construction. His most recent research publications include:
"The smallest interesting world?", "Babbage's
miraculous computation revisited," "Rulebound unruliness,"
"Renaissance mathematics and architectural proportion in
Alberti's De re aedificatoria," and "Architectonics
of proportion: a shape grammatical depiction of classical theory."
His book Architectonics of Humanism: Essays on Number in Architecture
before The First Moderns, a companion volume to Rudolf Wittkower's
Architectural Principles in the Age of Humanism was published,
together with a new edition of the Wittkower, in the Fall 1998.